Geodesic
Order-sharp estimates for Hardy-type operators on the cones of functions with properties of monotonicity
Eurasian mathematical journal, Tome 3 (2012) no. 2, pp. 53-84.

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Two-sided estimates are established for two types of generalized Hardy operators on the cones of functions in weighted Lebesgue spaces with some properties of monotonicity. We prove the results announced in [7], and present some other equivalent forms for the criterion of boundedness. Some other equivalent descriptions, particular cases, and results in the case of degenerate measures will be given in our next paper. 
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M. L. Goldman. Order-sharp estimates for Hardy-type operators on the cones of functions with properties of monotonicity. Eurasian mathematical journal, Tome 3 (2012) no. 2, pp. 53-84. http://geodesic.mathdoc.fr/item/EMJ_2012_3_2_a4/

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[2] V. I. Burenkov, M. L. Goldman, “Calculation of the norm of positive operator on the cone of monotone functions”, Proc. of the Steklov Inst. Math., 210, 1995, 65–89 | MR | Zbl

[3] M. Carro, A. Gogatishvili, J. Martin, L. Pick, “Weighted Inequalities Involving Two Hardy Operators with Applications to Embeddings of Function Spaces”, J. Operator Theory, 59:2 (2008), 309–332 | MR | Zbl

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[5] M. L. Goldman, “Sharp Estimates of the Norms of Hardy-Type Operators on the Cone of Quasimonotone Functions”, Proc. of the Steklov Inst. Math., 232, 2001, 109–137 | MR | Zbl

[6] M. L. Goldman, “On equivalent criteria for the boundedness of Hardy type operators on the cone of decreasing functions”, Analysis Mathematica, 37:2 (2011), 83–102 | DOI | MR | Zbl

[7] M. L. Goldman, “Order-sharp Estimates for Hardy-Type Operators on the Cones of Quasimonotone Functions”, Eurasian Mathematical Journal, 2:3 (2011), 143–146 | MR | Zbl

[8] M. L. Goldman, Hardy type inequalities on the cone of quasimonotone functions, Research Report no. 98/31, Russian Acad. Sci., Far-East Branch Computer Center, 1998